What't the best cruise for an older woman traveling alone if she doesn't want to share a cabin?

It kind of depends on what your budget it. Carnival os going to be the lowest. I think that Royal Caribbean is one of the better, low priced, cruise lines. In my opinion Royal Caribbean is a lot better than carnival for the price. Carnival is more for younger families and college age kids. So there is a lot of partying, drinking and gambling, if you aren't into this type stuff, you may want to stay away from Carnival.

Royal Caribbean, Norweigan, Princess, Crystal, Oceania, Disney are all great cruise lines.

Here is a website that gives lots of reviews on all the cruise lines, it is really helpful decide who you want to go with: http://cruisecritic.com/

Hope this helps